Theory | Linear temperaments | Just Approximation | Intervals | Compositions Other languages: Deutsch

Theory

The famous 53 equal division divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a 5-limit system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. In the 7-limit it tempers out 225/224, 1728/1715 and 3125/3087, the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out 99/98 and 121/120, and is the optimal patent val for Big Brother temperament, which tempers out both, as well as 11-limit orwell temperament, which also tempers out the 11-limit comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for athene temperament. It is the eighth zeta integral edo and the 16th prime edo, following 47edo and coming before 59edo.

53EDO has also found a certain dissemination as an EDO tuning for Arabic/Turkish/Persian music.

It can also be treated as a no-elevens, no-seventeens tuning, on which it is consistent all the way up to the 21-limit.

Wikipeda article about 53edo

Linear temperaments

List of edo-distinct 53et rank two temperaments

Just Approximation

53edo provides excellent approximations for the classic 5-limit just chords and scales, such as the Ptolemy-Zarlino "just major" scale.

interval	ratio	size	difference
perfect fifth	3/2	31	−0.07 cents
major third	5/4	17	−1.40 cents
minor third	6/5	14	+1.34 cents
major tone	9/8	9	−0.14 cents
minor tone	10/9	8	−1.27 cents
diat. semitone	16/15	5	+1.48 cents

One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.

The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the septimal kleisma, 225/224.

Intervals

degree	solfege	cents	approximate ratios	ups and downs notation			generator for
0	do	0.00	1/1	P1	unison	D
1	di	22.64	81/80, 64/63, 50/49	^1	up unison	D^
2	daw	45.28	49/48, 36/35, 33/32, 128/125	^^1, vvm2	double-up unison, double-down minor 2nd	D^^, Ebvv	Quartonic
3	ro	67.92	27/26, 26/25, 25/24, 22/21	vm2	downminor 2nd	Ebv
4	rih	90.57	21/20, 256/243	m2	minor 2nd	Eb
5	ra	113.21	16/15, 15/14	^m2	upminor 2nd	Eb^
6	ru	135.85	14/13, 13/12, 27/25	v~2	downmid 2nd	Eb^^
7	ruh	158.49	12/11, 11/10, 800/729	^~2	upmid 2nd	Evv	Hemikleismic
8	reh	181.13	10/9	vM2	downmajor 2nd	Ev
9	re	203.77	9/8	M2	major 2nd	E
10	ri	226.42	8/7, 256/225	^M2	upmajor 2nd	E^
11	raw	249.06	15/13, 144/125	^^M2, vvm3	double-up major 2nd, double-down minor 3rd	E^^, Fvv	Hemischis
12	ma	271.70	7/6, 75/64	vm3	downminor 3rd	Fv	Orwell
13	meh	294.34	13/11, 32/27	m3	minor 3rd	F
14	me	316.98	6/5	^m3	upminor 3rd	F^	Hanson/Catakleismic
15	mu	339.62	11/9, 243/200	v~3	downmid 3rd	F^^	Amity/Hitchcock
16	muh	362.26	16/13, 100/81	^~3	upmid 3rd	F#vv
17	mi	384.91	5/4	vM3	downmajor 3rd	F#v
18	maa	407.55	81/64	M3	major 3rd	F#
19	mo	430.19	9/7, 14/11	^M3	upmajor 3rd	F#^	Hamity
20	maw	452.83	13/10, 125/96	^^M3, vv4	double-up major 3rd, double-down 4th	F#^^, Gvv
21	fe	475.47	21/16, 675/512, 320/243	v4	down 4th	Gv	Vulture/Buzzard
22	fa	498.11	4/3	P4	perfect 4th	G
23	fih	520.75	27/20	^4	up 4th	G^
24	fu	543.40	11/8, 15/11	^^4	double-up 4th	G^^
25	fuh	566.04	18/13	vvA4, vd5	double-down aug 4th, downdim 5th	G#vv, Abv	Tricot
26	fi	588.68	7/5, 45/32	vA4, d5	downaug 4th, dim 5th	G#v, Ab
27	se	611.32	10/7, 64/45	A4, ^d5	aug 4th, updim 5th	G#, Ab^
28	suh	633.96	13/9	^A4, ^^d5	upaug 4th, double-up dim 5th	G#^, Ab^^
29	su	656.60	16/11, 22/15	vv5	double-down 5th	Avv
30	sih	679.25	40/27	v5	down 5th	Av
31	sol	701.89	3/2	P5	perfect 5th	A	Helmholtz/Garibaldi
32	si	724.53	32/21, 243/160, 1024/675	^5	up 5th	A^
33	saw	747.17	20/13, 192/125	^^5, vvm6	double-up 5th, double-down minor 6th	A^^, Bbvv
34	lo	769.81	14/9, 25/16, 11/7	vm6	downminor 6th	Bbv
35	leh	792.45	128/81	m6	minor 6th	Bb
36	le	815.09	8/5	^m6	upminor 6th	Bb^
37	lu	837.74	13/8, 81/50	v~6	downmid 6th	Bb^^
38	luh	860.38	18/11, 400/243	^~6	upmid 6th	Bvv
39	la	883.02	5/3	vM6	downmajor 6th	Bv
40	laa	905.66	22/13, 27/16	M6	major 6th	B
41	lo	928.30	12/7	^M6	upmajor 6th	B^
42	law	950.94	26/15, 125/72	^^M6 vvm7	double-up major 6th, double-down minor 7th	B^^, Cvv
43	ta	973.58	7/4	vm7	downminor 7th	Cv
44	teh	996.23	16/9	m7	minor 7th	C
45	te	1018.87	9/5	^m7	upminor 7th	C^
46	tu	1041.51	11/6, 20/11, 729/400	v~7	downmid 7th	C^^
47	tuh	1064.15	13/7, 24/13, 50/27	^~7	upmid 7th	C#vv
48	ti	1086.79	15/8	vM7	downmajor 7th	C#v
49	tih	1109.43	40/21, 243/128	M7	major 7th	C#
50	to	1132.08	48/25, 27/14	^M7	upmajor 7th	C#^
51	taw	1154.72	125/64	^^M7, vv8	double-up major 7th, double-down 8ve	C#^^, Dvv
52	da	1177.36	160/81	v8	down 8ve	Dv
53	do	1200	2/1	P8	perfect 8ve	D

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality	color	monzo format	examples
downminor	blue	{a, b, 0, 1}	7/6, 7/4
minor	fourthward white	{a, b}, b < -1	32/27, 16/9
upminor	green	{a, b, -1}	6/5, 9/5
downmid	jade	{a, b, 0, 0, 1}	11/9, 11/6
upmid	amber	{a, b, 0, 0, -1}	12/11, 18/11
downmajor	yellow	{a, b, 1}	5/4, 5/3
major	fifthward white	{a, b}, b > 1	9/8, 27/16
upmajor	red	{a, b, 0, -1}	9/7, 12/7

All 53edo chords can be named using ups and downs. Here are the blue, green, jade, yellow and red triads:

color of the 3rd	JI chord	notes as edosteps	notes of C chord	written name	spoken name
blue	6:7:9	0-12-31	C Ebv G	C.vm	C downminor
green	10:12:15	0-14-31	C Eb^ G	C.^m	C upminor
jade	18:22:27	0-15-31	C Eb^^ G	C.v~	C downmid
yellow	4:5:6	0-17-31	C Ev G	C.v	C downmajor or C dot down
red	14:18:27	0-19-31	C E^ G	C.^	C upmajor or C dot up